M(5,1,5)

M(5,1,5) - [math]B_0(kA_7)[/math]

Representative:	[math]B_0(kA_7)[/math]
Defect groups:	[math]C_5[/math]
Inertial quotients:	[math]C_4[/math]
[math]k(B)=[/math]	5
[math]l(B)=[/math]	4
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix:	[math]\left( \begin{array}{cccc} 2 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 2 \\ \end{array} \right)[/math]
Defect group Morita invariant?	Yes
Inertial quotient Morita invariant?	Yes
[math]\mathcal{O}[/math]-Morita classes known?	Yes
[math]\mathcal{O}[/math]-Morita classes:	[math]B_0(\mathcal{O}A_7)[/math]
Decomposition matrices:	[math]\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math]	1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math]
[math]PI(B)=[/math]	{{{PIgroup}}}
Source algebras known?	Yes
Source algebra reps:
[math]k[/math]-derived equiv. classes known?	Yes
[math]k[/math]-derived equivalent to:	M(5,1,4), M(5,1,6)
[math]\mathcal{O}[/math]-derived equiv. classes known?	Yes
[math]p'[/math]-index covering blocks:	{{{coveringblocks}}}
[math]p'[/math]-index covered blocks:	{{{coveredblocks}}}
Index [math]p[/math] covering blocks:	{{{pcoveringblocks}}}

Basic algebra

Quiver: a:<1,2>, b:<2,3>, c:<3,4>, d:<4,3>, e:<3,2>, f:<2,1>

Relations w.r.t. [math]k[/math]: ab=bc=de=ef=0, fa=be, eb=cd

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, S_2, S_3, S_4[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{cccc} \begin{array}{c} S_1 \\ S_2 \\ S_1 \\ \end{array}, & \begin{array}{ccc} & S_2 & \\ S_1 & & S_3 \\ & S_2 & \\ \end{array}, & \begin{array}{ccc} & S_3 & \\ S_2 & & S_4 \\ & S_3 & \\ \end{array}, & \begin{array}{c} S_4 \\ S_3 \\ S_4 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math]C_5[/math]